Factual error: When Edward Teller proposed that a fission explosion might ignite the atmosphere in an uncontrollable chain reaction, Oppenheimer did not go to New Jersey to consult with Einstein. He actually went to Michigan to consult with experimentalist Arthur H. Compton.
Factual error: The real Edward Teller walked with a pronounced limp due to his right foot being almost entirely severed in a streetcar accident in 1928. In the film, he walks normally.
Factual error: No one would have used the phrase "black hole" in 1939. The term "black hole" was first used in 1963 in "Life" and "Science News" and by Ann Ewing in an article in January of 1964. Princeton physicist John Wheeler popularised the term.
Suggested correction: Nobody uses the term "black hole" in the movie, only the term "dark star". Oppenheimer once refers to it as a hole in space, but not a black hole.
When Oppenheimer walks into the room of cheering people (after he says he'll be in Pasadena), someone says "paper on black holes, it's in!"
Ah, yes, I see. I wonder, though, if it's really that unlikely someone would call it a black hole before it was popularized? It is essentially what they are. Certainly, it's possible somebody before 1963 called it that without it ending up in a paper. Just a coincidence, then.
Continuity mistake: When Oppenheimer and Kitty are on the ranch, drinking from the flask in turns, he takes a sip. But from a different angle, his hand is suddenly down.
Continuity mistake: When Los Álamos is being built, Oppenheimer talks with a guy and draws four letters and circles on the board behind. Their styles change between takes.
Continuity mistake: When Oppenheimer comforts Jean, the position of his hand on her forehead keeps changing in every shot.
Continuity mistake: After Oppenheimer meets Jean, he takes the glass to his mouth, but from a different angle, it's lowered.
Continuity mistake: When Oppenheimer meets Jean, she is staring at him. Angle changes, and she is looking downward. Back to the first angle, and she stares at him again.